Lecturer Splitting bundles

e Recap A basefield K X Y conical symplic reselinR Atumaya algebra on X

Thm Lec 3 St R Ifi H XR a f i oIii Sthasfinite homologicaldimension

Then RT D CehR t D Stmod

1 Derived equivalences fromquantinsObservation If D is Frobenius constantfilteredquantinof X then it's Azumaya algebra on X

Example X T GB FF char Fp Then Dem isAzumayaalgebra on X

P 1 Condition i Suppose 04 Xa Ya is conicalsymplicresolution over Q o Xp Ya is definedover a finite loanof TL denotedby R For pro alg closedfield Fofcharp a si X Yg a conical symplectic resolution

Thm Let D filteredFrobenius constantquantin ofX so

Azumayaalgebra on X ThenHills D so tire

ProofSteph Claim H Xp 0 o fire

i ti Xp 9 so t i o this is a special caseofGranert Riemenschneider theorem if 14 Xe Ya a birational

projective morphism Xa is smooth then Ri Kygo tireafterfinite localization

ii ti Xr O off i o by i H xpO is a torsion

Rmodule Since IT isprojective HxpO isfinitelygener'dever RLY But RLY isfinitelygenerated R algebra SoH XrO is killed by inverting finitelymanyprimes

iii Have exact sequence O O'xp O'xp xg.peApply long exact sequence in cohomology

Hi App 0 03 f i a Hixp d so fine

Step2 Fr X X's isfinite ti XsFredH XgOx o f i oRecall fromLec 3 have an F equiv't coherentsheaf

DE on XI xSpec FCh s tDf hDt t FriOx a

Df h 1 Dit IDEnough to show H XpSpecFct Dt o f i o

By longexact sequence for aI

IHiDat HiDat ti fr Ox 03 2

finitelygeneratedgraded FLY h FExactmodule But the algebra ispositivelygradedBygradedNakayama 2 3Hi Df e fine a

Exercise Prove that He X D is a filteredquantizationofFLY E F xD

12 Finite homological dimension At least inexamples wecan find D s t H T D have finite homological dimensionSometimes X T GB X resolutionofsymplequotientsinglyeg X Hilton F can findsuch D directly In othercases can argue by reduction from char0Quantizations of Xp classifiedby H X E e

For Da corresp to a Zariskigeneric Rett X E Dayhave finite homological dimension For TEH XA canreduce

Dey T Day medp so in all examples thereductionof

Day is Frobenius constant One can essentially showthatreducing Dey modpro preserves thehomological

dimension

Conclusion Essentially always can findFrobeniusconstquantinD s t RT D CahD I D CAmod 54 11011

But how useful isthis Often we care about St For T GBis a HarishChandra central redin ofUlog CahD not

so much

On the otherhand if Azumaya algebra R that issplitR End V where V is vector bundle thenCahRig CehX

Issue a Frobenius constant quantization doesn'tsplit

Fix Still D splits somewhere thisallows toproduce a

split Azumaya algebra on XI andevenXe that willproduce a derivedequivalence

2 Splitting bundlesReminder if E isany variety ZE E Qq completelocal

rings SpecOz Restriction ofany Azumayaalgebrato this subscheme splits

Fact In cases of interest the restriction ofFrobeniusconstant quantization D to

yo s xyo X s N

splits t yet ysmallneighd of fiber ofy

Thisfellows from combining KubratTrarkin Bogdanova

Vologedsky for certain Frobeniusconstantquantins that inexamples are reducedmodp from char l and so include DW D has finite homological dimension

Take y o X Let EI be a splittingbundlefor lycandefinedup to twisting w line bundle EndEjI s D formalfunctionthm D ya n EndEfExti EgE's H XD seRT D CahD D ImadRT D CehDlxc.sn I D A mod

f EgoD CahX n

2 1 Extension to XS Want toextendE's to a vectorbundleover Xp5 AXp f aXf's F a Xg

Fact Velegedsky Since Ext E'sEf 03E'shas anFSquirt structure

We'll use this to extend Ef to Xp Recall haveprey'vereselin morphism it XI Y F equivit FLY is

positively graded equivalently F contracts X to lol

5 A E's F a St EndEfA 5 finitepart of It gradedalgebraover F Y

Exercise Thegrading on Ag is boundedfrombelow

AT St

Completionfunctor A mod Atm L

Fact Restriction to X defines an equivalenceCeh XI Cah gin

EjLet EgeCeh XI bethe image of Ejunder the equivalence

Exercise End Eg Is A Ext EgEg o fine

Lemma RT EgO D CahX l ID AgmodProofRile jiced x D St'smod

RT EGO D Cah x D f'smodeI 1

RICE Ox D Cah x I D I mode

Rf Ego D Cah x D SImod I

2 2 Lift to characteristic lXY Il so we can view E as a vectorbundle overXpIt is defined over Eg Eg ever Xp Let R be an algaextension of TL sit

R FX is defined over R and isnice

Xp is a closedsubscheme ofXp

Let 12 9 completion of R at Ker R sEgConsider Xpt formalneigh'd ofXp inXp a formalscheme

Since Ext Egg Egg o for is 1,2 Egg deforms uniquely toa vector bundle over Xp the deformin is GmequivitSince Gmaction is contracting we can algebrite this

deformin getting an equiv't vector bundle onXpng

R Gene get vector bundle Ee en te

Properties It's a equivitDesEnd Ee hasfinite homologicaldiminExt EeEe so tireRICE D ohYe ID Agmod

23 Identifying AeNotice Ee depends onp rocEesp

him Howeverbypickingdirect summand of Ee w diff it multiplicities in knownexamples can achieve that Ed is independentofpCan describe Te inthefollowing casesI Y VIT V is sympl vectorspace Esp V finitegrip

Then DeCEV TII X is smooth Coulomb branch of agaugethey

constructed BFN Ae was describedbyWebster Possible Xinclude hypertonic var's finite affinetype A Nakajimaquivervar's

Fact Kaledin Aa depends only on Y butnotonXFor two symplic reselins XX of Y have

D CahX ID Domed D CahXThis is a special case of Kequivalence D equivalenceconjecture